COMPARISON OF PRECONDITIONED GENERALIZED CONJUGATE-GRADIENT METHODS TO 2-DIMENSIONAL NEUTRON AND PHOTON TRANSPORT-EQUATION

We apply and compare the preconditioned generalized conjugate gradient methods to solve the linear system equation that arises in the two-di mensional neutron and photon transport equation in this paper. Several subroutines are developed on the basis of preconditioned generalized conjugate gradient methods for time-independent, two-dimensional neutr on and photon transport equation in the transport theory. These genera lized conjugate gradient methods are used: TFQMR (transpose free quasi -minimal residual algorithm) CGS (conjugate gradient square algorithm) , Bi-CGSTAB (biconjugate gradient stabilized algorithm) and QMRCGSTAB (quasi-minimal residual variant of bi-conjugate gradient stabilized al gorithm). These subroutines are connected to computer program DORT. Se veral problems are tested on a personal computer with Intel Pentium CP U. The reasons to choose the generalized conjugate gradient methods ar e that the methods have better residual (equivalent to error) control procedures in the computation and have better convergent rate. The poi ntwise incomplete LU factorization ILU, modified pointwise incomplete LU factorization MILU, block incomplete factorization BILU and modifie d blockwise incomplete LU factorization MBILU are the preconditioning techniques used in the several testing problems, In Bi-CGSTAB, CGS, TF QMR and QMRCGSTAB method, we find that either CGS or Bi-CGSTAB method combined with preconditioner MBILU is the most efficient algorithm in these methods in the several testing problems. The numerical solution of flux by preconditioned CGS and Bi-CGSTAB methods has the same resul t as those from Gray computer, obtained by either the point successive relaxation method or the line successive relaxation method combined w ith Gaussian elimination, (C) 1997 Elsevier Science Ltd.